I can't really tell what's wrong from the pictures, but I can offer some advice to help diagnose the problem.
First, I understand from the question that all of the notes with dampers are affected, so the piano is behaving as if you're constantly stepping on the pedal. If that's not correct, stop reading and leave a comment explaining which notes are affected.
Second, if you're uncomfortable unscrewing and removing the pedal assembly and then replacing it then you should seek assistance from a handy friend, or just bite the bullet and call the piano tuner. The piano surely needs tuning anyway, and fixing the dampers is probably just a question of realigning something that is out of alignment.
Third, a likely cause for this condition is indeed that the pedal is assembled incorrectly. What does happen when you step on the pedal? Is everything moving? Does the pedal move as it normally does, along with the vertical rod and the lever shown in the picture? If they are not moving normally, try unscrewing the pedal assembly and moving the lever manually. If the dampers now work as expected, you can probably just put the pedal assembly back together carefully and everything should work as normal. Be certain to check that the vertical rod also slides relatively freely before you put everything back together.
Finally, if you remove the pedal assembly and find that manipulating the lever directly does not solve the problem, then it's likely that some lever or rod has fallen out of alignment during the move. (For example, the vertical wooden rod that extends upward from the wooden lever through the hole in the piano case may have fallen out of a socket at the other end, holding that piece in a higher position than it should and therefore keeping the dampers off of the strings.) If that is the case then it will probably be necessary to remove the action (keys and connected parts) to get at the internal mechanism. This isn't as hard as it sounds, but it's also not trivial. If you're a confident tinkerer, you might give it a go, but, once again, if you're planning to get the instrument tuned anyway, just go ahead and do that. The technician isn't likely to charge much to realign the mechanism.